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T-TEST
Outline
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Introduction
T Distribution
Example cases
Test of Means-Single population
Test of difference of Means-Independent Samples
Test of difference of Means-Independent Samples
Practical cases on SPSS
Rahul Chandra
T-Test
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The t-test was introduced in 1908 by William Sealy
Gosset, a chemist working for the Guiness Brewery in
Dublin, Ireland to monitor the quality of stout – a dark
beer.
Because his employer did not want to reveal the fact
that it was using statistics for quality control, Gosset
published the test in Biometrika using his pen name
“Student” (he was a student of Sir Ronald Fisher), and
the test involved calculating the value of t.
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Hypothesis Formulation
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Hypotheses are a pair of mutually exclusive,
collectively exhaustive statements about the world.
One statement or the other must be true, but they
cannot both be true.
H0: Null Hypothesis
H1: Alternative Hypothesis
These two statements are hypotheses because the truth
is unknown.
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Logic of Hypothesis Testing
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Efforts will be made to reject the null hypothesis.
If H0 is rejected, we tentatively conclude H1 to be
the case.
H0 is sometimes called the maintained hypothesis.
H1 is called the action alternative because action
may be required if we reject H0 in favor of H1.
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Logic of Hypothesis Testing
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Can Hypotheses be Proved?
We cannot prove a null hypothesis, we can only fail
to reject it.
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Role of Evidence
The null hypothesis is assumed true and a
contradiction is sought.
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Sampling distribution
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It is the theoretical distribution of an infinite
number of samples from the population of interest
in your study.
It may distribute any sample statistic like mean, sum,
proportions etc. For two population it can be of
difference of means, etc.
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Standard error
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Every sample always has some inherent level of
error, called the standard error. It is basically the
standard deviation of sampling distribution.
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Central Limit Theorem
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It states that even if the population distribution is
not normal, sampling distribution can still be normal
provided the sample size is large enough (> 30).
For small samples normality condition does not hold.
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T-Distribution
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In case the sample size is small (n ≤ 30) and is drawn from
a normal population with unknown standard deviation σ, a
t test is used to conduct the hypothesis for the test of mean.
The t distribution is a symmetrical distribution just like the
normal one.
However, t distribution is higher at the tail and lower at the
peak. The t distribution is flatter than the normal
distribution.
With an increase in the sample size (and hence degrees of
freedom), t distribution loses its flatness and approaches
the normal distribution whenever n > 30.
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T-Distribution
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A comparative shape of t and normal distribution is given in the
figure below:
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Assumptions
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Test Concerning Means – Case of
Single Population
Case of large sample - In case the sample size n is
large or small but the value of the population standard
deviation is known, a Z test is appropriate. The test
statistic is given by,
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Testing a Mean:
Known Population Variance
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Step 1: State the hypotheses
For example, H0: m < 216
H1: m > 216
Step 2: Specify the decision rule
For example, for a = .05
for the right-tail area,
Reject H0 if z > 1.645,
otherwise do not reject H0
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Testing a Mean:
Known Population Variance
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For a two-tailed test, we split the risk of Type I
error by putting a/2 in each tail.
For example, for a = .05
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Test Concerning Means – Case of
Single Population
If the population standard deviation σ is unknown, the
sample standard deviation
is used as an estimate of σ. There can be alternate
cases of two-tailed and one-tailed tests of hypotheses.
Corresponding to the null hypothesis H0 : μ = μ0, the
following criteria could be formulated as shown in the
table below:
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Test Concerning Means – Case of
Single Population
The null hypothesis to be tested is:
H0 : μ = μ0
The alternative hypothesis could be one-tailed or two-tailed test.
The test statistics used in this case is:
The procedure for testing the hypothesis of a mean is
identical to the case of large sample.
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Tests for Difference Between Two
Population Means
Case of large sample - In case both the sample sizes are
greater than 30, a Z test is used. The hypothesis to be tested
may be written as:
H0 : μ1 = μ2
H1 : μ1 ≠ μ2
Where,
μ1 = mean of population 1
μ2 = mean of population 2
The above is a case of two-tailed test. The test statistic used is
on the
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Tests for Difference Between Two
Population Means
The Z value for the problem can be computed using the above formula and
compared with the table value to either accept or reject the hypothesis.
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Tests for Difference Between Two
Population Means
Case of small sample - If the size of both the samples
is less than 30 and the population standard deviation is
unknown, the procedure described above to discuss the
equality of two population means is not applicable in
the sense that a t test would be applicable under the
assumptions:
a) Two population variances are equal.
b) Two population variances are not equal.
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Tests for Difference Between Two
Population Means
Population variances are equal - If the two population variances are
equal, it implies that their respective unbiased estimates are also equal.
In such a case, the expression becomes:
To get an estimate of σˆ2, a weighted average of s12 and s22 is used, where
the weights are the number of degrees of freedom of each sample. The
weighted average is called a ‘pooled estimate’ of σ2. This pooled estimate is
given by the expression:
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Tests for Difference Between Two
Population Means
The testing procedure could be explained as under:
H0 : μ1 = μ2 ⇒ μ1 – μ2 = 0
H1 : μ1 ≠ μ2 ⇒ μ1 – μ2 ≠ 0
In this case, the test statistic t is given by the expression:
Once the value of t statistic is computed from the sample data, it is
compared with the tabulated value at a level of significance α to
arrive at a decision regarding the acceptance or rejection of
hypothesis.
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Tests for Difference Between Two
Population Means
Population variances are not equal - In case population variances
are not equal, the test statistic for testing the equality of two population
means when the size of samples are small is given by:
The degrees of freedom in such a case is given by the expression:
The procedure for testing of hypothesis remains the same as was discussed
when the variances of two populations were assumed to be same.
Rahul Chandra